In many real-time and embedded systems, suspension delays may occur when tasks block to access shared resources or interact with external devices. Unfortunately, prior analysis methods for dealing with suspensions are quite pessimistic. In this paper, a novel technique is presented for analyzing soft real-time sporadic self-suspending task systems, for which bounded deadline tardiness is required, scheduled under global schedulers such as global EDF on multiprocessors (or EDF on uniprocessors). This technique is used to derive a new schedulability test that results in only O(m) suspension-related utilization loss, where m is the number of processors.